MATH - BIOINF - STATS 547: Mathematics of Data

Instructor: Prof. INDIKA Rajapakse (

Teaching Assistant: COOPER Stansbury (

Class Time: Tuesday and Thursday, 2:30 - 4:00PM in EH B844:

Recorded Lectures: Available on Canvas only

Office Hours:

Timeline: Link to Timeline


Final Class Day Resources:


This section contains class notes as well as helpful resources, including classic papers, grouped by topic. Please browse this section, and keep it in mind in your future work beyond this course.

Singular Value Decomposition (SVD)

Eckhart-Young Theorem and Low Rank Approximations

Proof (Book: Golub and Van Loan, Matrix computations )

  • Additional Reading

  1. Gavish, Matan, and David L. Donoho. "The optimal hard threshold for singular values is 4/sqrt(3)." IEEE Transactions on Information Theory 60.8 (2014): 5040-5053. (Amazing Paper!)

  2. Udell, Madeleine, and Alex Townsend. "Why are big data matrices approximately low rank?." SIAM Journal on Mathematics of Data Science 1.1 (2019): 144-160.

  3. Turk, Matthew, and Alex Pentland. "Eigenfaces for recognition." Journal of cognitive neuroscience 3.1 (1991): 71-86. (Classic! just browse)

Dimension Reduction

Linear Methods

Nonlinear Methods


Remark: This is a really nice review in general, but the most relevant parts are pages 9 and 10

The Laplacian


  1. Ng, Andrew Y., Michael I. Jordan, and Yair Weiss. "On spectral clustering: Analysis and an algorithm." Advances in neural information processing systems 2 (2002): 849-856.

  2. Belkin, Mikhail, and Partha Niyogi. "Laplacian eigenmaps and spectral techniques for embedding and clustering." Advances in neural information processing systems. 2002.

  3. Von Luxburg, Ulrike. "A tutorial on spectral clustering." Statistics and computing 17.4 (2007): 395-416. (Excellent Review!)

  4. Cheeger J. A lower bound for the smallest eigenvalue of the Laplacian. In Problems in analysis 2015 Mar 8 (pp. 195-200). Princeton University Press.

  5. Chen, Pin-Yu, et al. "Fast incremental von neumann graph entropy computation: Theory, algorithm, and applications." International Conference on Machine Learning. PMLR, 2019.

The Turing System


  1. Turing, Alan. "The chemical basis of morphogenesis." Philosophical Transactions of the Royal Society of London B. 237 (1952): 641

  2. Rajapakse I, and Smale S. "Emergence of Function from Coordinated Cells in a Tissue." Proceedings of the National Academy of Sciences 114.7 (2017): 1462-1467.

Dynamic Mode Decomposition (DMD)


  • Kutz, J. Nathan, et al. Dynamic mode decomposition: data-driven modeling of complex systems. Society for Industrial and Applied Mathematics, 2016.

Chapter 1: Dynamic Mode Decomposition: An Introduction


  1. Hirsh, Seth M., et al. "Centering data improves the dynamic mode decomposition." SIAM Journal on Applied Dynamical Systems 19.3 (2020): 1920-1955.

  2. T. Askham, P. Zheng, A. Aravkin, and J. N. Kutz, Robust and Scalable Methods for the Dynamic Mode Decomposition, SIAM J. Appl. Dyn. Syst., vol. 21, no. 1, pp. 60–79, Mar. 2022

  3. P. J. Baddoo, B. Herrmann, B. J. McKeon, J. N. Kutz, and S. L. Brunton, Physics-informed dynamic mode decomposition (piDMD),Dec. 2021

  4. Z. Bai, E. Kaiser, J. L. Proctor, J. N. Kutz, and S. L. Brunton, Dynamic Mode Decomposition for Compressive System Identification,” AIAA Journal, vol. 58, no. 2, pp. 561–574, 2020

  5. J. J. Bramburger, D. Dylewsky, and J. N. Kutz, Sparse identification of slow timescale dynamics, Phys. Rev. E, vol. 102, no. 2, p. 022204, Aug. 2020

  6. S. L. Brunton and J. N. Kutz, Methods for data-driven multiscale model discovery for materials, J. Phys. Mater., vol. 2, no. 4, p. 044002, Jul. 2019

  7. C. W. Curtis and D. J. Alford-Lago, Dynamic-mode decomposition and optimal prediction, Phys. Rev. E, vol. 103, no. 1, p. 012201, Jan. 2021

  8. U. Fasel, E. Kaiser, J. N. Kutz, B. W. Brunton, and S. L. Brunton, SINDy with Control: A Tutorial, in 2021 60th IEEE Conference on Decision and Control (CDC), Dec. 2021

  9. M. R. Jovanović, P. J. Schmid, and J. W. Nichols, Sparsity-promoting dynamic mode decomposition, Physics of Fluids, vol. 26, no. 2, p. 024103, Feb. 2014

  10. N. M. M. Kalimullah, A. Shelke, and A. Habib, Multiresolution Dynamic Mode Decomposition (mrDMD) of Elastic Waves for Damage Localisation in Piezoelectric Ceramic, IEEE Access, vol. 9, pp. 120512–120524, 2021

  11. S. Klus, P. Gelß, S. Peitz, and C. Schütte, Tensor-based dynamic mode decomposition, Nonlinearity, vol. 31, no. 7, pp. 3359–3380, Jun. 2018

  12. J. N. Kutz, X. Fu, S. L. Brunton, and N. B. Erichson, Multi-resolution Dynamic Mode Decomposition for Foreground/Background Separation and Object Tracking, in 2015 IEEE International Conference on Computer Vision Workshop (ICCVW), Dec. 2015

  13. J. N. Kutz, X. Fu, and S. L. Brunton, Multiresolution Dynamic Mode Decomposition, SIAM J. Appl. Dyn. Syst., vol. 15, no. 2, pp. 713–735, Jan. 2016

  14. S. Le Clainche and J. M. Vega, Higher Order Dynamic Mode Decomposition, SIAM J. Appl. Dyn. Syst., vol. 16, no. 2, pp. 882–925, Jan. 2017

  15. I. Mezić, Spectral Properties of Dynamical Systems, Model Reduction and Decompositions, Nonlinear Dyn, vol. 41, no. 1, pp. 309–325, Aug. 2005

  16. P. J. Schmid, Dynamic mode decomposition of numerical and experimental data, Journal of Fluid Mechanics, vol. 656, pp. 5–28, Aug. 2010

  17. J. H. Tu, C. W. Rowley, D. M. Luchtenburg, S. L. Brunton, and J. N. Kutz, On Dynamic Mode Decomposition: Theory and Applications, Journal of Computational Dynamics, vol. 1, no. 2, pp. 391–421, 2014

Koopman Theory

Koopman Analysis


  1. Mezic I. Koopman operator, geometry, and learning. arXiv preprint arXiv:2010.05377. 2020 Oct 12.

  2. Champion K, Lusch B, Kutz JN, Brunton SL. "Data-driven discovery of coordinates and governing equations." Proceedings of the National Academy of Sciences. 2019 Nov 5;116(45):22445-51.

  3. Surana A. Koopman operator framework for time series modeling and analysis. Journal of Nonlinear Science. 2020 Oct;30(5):1973-2006.

Video: Koopman Operator Theory for Dynamical Systems, Control and Data Analytics by Prof. Igor Mezic

Data Guided Control (DGC)

  1. DGC

  2. Controllability

  3. Network Controllability (example)


  1. Ronquist S, Patterson G, Muir LA, Lindsly S, Chen H, Brown M, Wicha M, Bloch A, Brockett R and Rajapakse I. "Algorithm for Cellular Reprogramming." Proceedings of the National Academy of Sciences 114.45 (2017): 11832-11837.

  2. Moore B. Principal component analysis in linear systems: Controllability, observability, and model reduction. IEEE transactions on automatic control. 1981 Feb;26(1):17-32. (Amazing Paper!)

PagRank (PR)

  1. PR


  1. Bryan, Kurt, and Tanya Leise. "The $25,000,000,000 eigenvector: The linear algebra behind Google." SIAM review 48.3 (2006): 569-581.

Causal Discovery from Data: Dr. David Heckerman

  1. Slides

Matrix Completion and CVX

  1. Matrix Completion in the Nuclear Norm

  2. Convex Optimization Tools: CVX

  3. Mathematics of Optimization

  4. Robust PCA


  1. Candès, Emmanuel J., et al. "Robust principal component analysis?." Journal of the ACM (JACM) 58.3 (2011): 1-37.

  2. Scherl I, Strom B, Shang JK, Williams O, Polagye BL, Brunton SL. Robust principal component analysis for modal decomposition of corrupt fluid flows. Physical Review Fluids. 2020 May 28;5(5):054401.

  3. Becker SR, Candès EJ, Grant MC. Templates for convex cone problems with applications to sparse signal recovery. Mathematical programming computation. 2011 Sep;3(3):165-218.

  4. Koren Y, Bell R, Volinsky C. Matrix factorization techniques for recommender systems. Computer. 2009 Aug 7;42(8):30-7.

  5. Van Dijk D, Sharma R, Nainys J, Yim K, Kathail P, Carr AJ, Burdziak C, Moon KR, Chaffer CL, Pattabiraman D, Bierie B. Recovering gene interactions from single-cell data using data diffusion. Cell. 2018 Jul 26;174(3):716-29.

Compressive Sensing

  1. "Magic" Reconstruction: Compressed Sensing (MATLAB Code)

  2. Backslash

  3. The following two chapters are good references on Compressive Sensing [2, 3].

Sparsity and Compressed Sensing

Basics of Compressed Sensing

  1. Terence Tao's summary of the current state of compressive sampling theory


  1. Candès EJ, Wakin MB. An introduction to compressive sampling. IEEE signal processing magazine. 2008 Mar 21;25(2):21-30. (Excellent Review!)

  2. Brunton SL, Kutz JN. Data-driven science and engineering: Machine learning, dynamical systems, and control. Cambridge University Press; 2019 Feb 28.

  3. Kutz JN. Data-driven modeling & scientific computation: methods for complex systems & big data. Oxford University Press; 2013 Aug 8.

Tensors and Hypergraphs

  1. Introductory Slides

  2. Tensors and Hypergraphs

Additional Slides: AFOSR 2021 slides

  1. Hypergraph Entropy and Hypergraph Distance Measures

  2. Multi-Correlation Summary

  3. Decomposing a Tensor: Examples


TensorLy Python Toolbox

Hypergraph Visualization: PAOHvis


  • Eldén, Lars. Matrix methods in data mining and pattern recognition. Society for Industrial and Applied Mathematics, 2007.

Chapter 8: Tensor Decomposition


Nonnegative Matrix Factorization


  1. Hessian Matrices and Gradient Descent


  1. Lee D, Seung HS. Algorithms for non-negative matrix factorization. Advances in neural information processing systems. 2000;13.

  2. Lee, Daniel D., and H. Sebastian Seung. "Learning the parts of objects by non-negative matrix factorization." Nature 401.6755 (1999): 788-791.


  1. Cichocki A, Zdunek R, Phan AH, Amari SI. Nonnegative matrix and tensor factorizations: applications to exploratory multi-way data analysis and blind source separation. John Wiley & Sons; 2009 Jul 10.

MATLAB Image Decomposition Examples

  1. Codes

TED Talk: I am my connectome

Topological Data Analysis

  1. A Taste of Topology: Book Chapter: Charles Pugh. Real mathematical analysis. (Excellent Exposition!)

  2. Morse Theory

  3. Introduction: These slides deck includes slides from my good friend Yuan Yao


  1. Topology ToolKit

WEB Resources

  1. Persistent Homology

  2. GDA


  1. Wasserman L. Topological data analysis. Annual Review of Statistics and Its Application. 2018 Mar 7;5:501-32.

  2. Tierny, J. (2017). Introduction to topological data analysis

  3. Nicolau M, Levine AJ, Carlsson G. Topology based data analysis identifies a subgroup of breast cancers with a unique mutational profile and excellent survival Proceedings of the National Academy of Sciences. 2011 Apr 26;108(17):7265-70.

  4. Carlsson G. Topology and Data. Bulletin of the American Mathematical Society. 2009;46(2):255-308.

  5. Lum PY, Singh G, Lehman A, Ishkanov T, Vejdemo-Johansson M, Alagappan M, Carlsson J, Carlsson G. Extracting insights from the shape of complex data using topology. Scientific reports. 2013 Feb 7;3(1):1-8.


  1. Biological data

  2. Course Introduction Data

  3. Austin Benson Datasets (Hypergraph Data)

  4. Metabolite (Hypergraph Data)

  5. COVID data: Slides from Dr. Dennis Chao from Gates Foundation

DATA Visualization


I will add to this list throughout the semester

  1. Donoho D. "50 years of data science." Journal of Computational and Graphical Statistics. 2017 Oct 2;26(4):745-66.

  2. Rajapakse, Indika. "Conversation with Dr. Steve Smale and Dr. Lee Hartwell." NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY 68, no. 9.

  3. The Nobel Prize in Physiology or Medicine 2012

  4. Aksoy SG, Hagberg A, Joslyn CA, Kay B, Purvine E, Young SJ. Models and Methods for Sparse (Hyper) Network Science in Business, Industry, and Government. NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY.;69(2).

  5. Smale, Steve. "Mathematical problems for the next century." The mathematical intelligencer 20.2 (1998): 7-15.

  6. Kolda T. Mathematics: The Tao of Data Science. (2020).